Inner measure


In mathematics, in particular in measure theory, an inner measure is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set.

Definition

An inner measure is a function
defined on all subsets of a set X, that satisfies the following conditions:
Let Σ be a σ-algebra over a set X and μ be a measure on Σ.
Then the inner measure μ* induced by μ is defined by
Essentially μ* gives a lower bound of the size of any set by ensuring it is at least as big as the μ-measure of any of its Σ-measurable subsets. Even though the set function μ* is usually not a measure, μ* shares the following properties with measures:

Measure completion

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If μ is a finite measure defined on a σ-algebra Σ over X and μ* and μ* are corresponding induced outer and inner measures, then the sets T ∈ 2X such that μ* = μ* form a σ-algebra with. The set function μ̂ defined by
for all is a measure on known as the completion of μ.